Marble on a track, reaching square 12 first

Question

A marble rests on square 3 of a numbered track running from square 0 to square 12. Each move it slides to an adjacent square: one step right with probability 	frac12, one step left with probability 	frac12. It stops the instant it lands on square 12 (the prize end) or square 0 (the drain). What is the probability the marble reaches square 12 before square 0?

Accepted Answer

Let pk be the probability of hitting 12 before 0 from square k, with p0 = 0 and p{12} = 1. Each move is a fair coin flip, so first-step analysis gives pk = 	frac12 p{k-1} + 	frac12 p{k+1} \quad\Longrightarrow\quad p{k+1} - pk = pk - p{k-1}. Equal gaps mean pk is linear in k. Fitting the ends p0 = 0 and p{12} = 1: pk = \frac{k}{12}, \qquad p3 = \frac{3}{12} = \frac14 = 0.25. Takeaway. For a symmetric walk between 0 and N, the probability of reaching N first is k/N, the fraction of the way you already stand. The barrier positions, not the step size, fix the answer.

Marble on a track, reaching square 12 first

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